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Cumulants of a random variable distributed uniformly on the first \(n\) integers

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Abstract

Motivated by a roundoff problem, we derive new expressions for cumulants of a random variable distributed uniformly on \(0,1, \ldots , n-1\). Their computational efficiency over a known expression is discussed.

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References

  • Abramowitz, M., Stegun, I.A.: (1964) Handbook of mathematical functions. United States Department of Commerce, National Bureau of Standards, Applied Mathematics Series, vol. 55

  • Comtet, L.: Advanced combinatorics. Reidel, Dordrecht (1974)

    Book  MATH  Google Scholar 

  • Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series and products, 6th edn. Academic Press, New York (2000)

    Google Scholar 

  • Hoh, J.-K., Cha, K.-J., Park, M.-I., Lee, M.-L.T., Park, Y.-S.: Estimating time to full uterine cervical dilation using genetic algorithm. Kaohsiung J. Med. Sci. 28, 423–428 (2012)

    Article  Google Scholar 

  • Johnson, N.L., Kotz, S., Kemp, A.W.: Univariate discrete distributions, 2nd edn. Wiley, New York (1992)

    MATH  Google Scholar 

  • Kendall, M.G.: Conditions under which Sheppard’s corrections are valid. J. R. Stat. Soc. 101, 592–605 (1938)

    Article  Google Scholar 

  • Kendall, M.G., Stuart, A.: The advanced theory of statistics, vol. 1, 4th edn. Macmillan Publishing Company, New York (1979)

  • Lee, C.S., Vardeman, S.B.: Interval estimation of a normal process standard deviation from rounded data. Commun. Stat. Simul. Comput. 31, 13–34 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Lee, C.S., Vardeman, S.B.: Confidence intervals based on rounded data from the balanced one-way normal random effects model. Commun. Stat. Simul. Comput. 32, 835–856 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Li, W., Bai, Z.D.: Analysis of accumulated rounding errors in autoregressive processes. J. Time Ser. Anal. 32, 518–530 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Schneeweiss, H., Komlos, J.H., Ahmad, A.: Symmetric and asymmetric rounding: a review and some new results. AStA Adv. Stat. Anal. 94, 247–271 (2010)

    Article  MathSciNet  Google Scholar 

  • Sheppard, W.F.: Central-difference formulae. Proc. Lond. Math. Soc. 31, 449–488 (1899)

    Article  MATH  MathSciNet  Google Scholar 

  • Sheppard, W.F.: The calculation of moments of a frequency distribution. Biometrika 5, 450–459 (1907)

    Article  Google Scholar 

  • Thorarinsdottir, T.L., Johnson, M.S.: (2012) Probabilistic wind gust forecasting using nonhomogeneous Gaussian regression. Monthly Weather Review, pp. 889–897

  • Wang, A., Wertelecki, W.: Density estimation for data with rounding errors. Comput. Stat. Data Anal. 65, 4–12 (2013)

    Article  MathSciNet  Google Scholar 

  • Withers, C.S., Nadarajah, S.: Unbiased estimates for linear regression with roundoff error. Prob. Math. Stat. 31, 177–182 (2011)

    MATH  MathSciNet  Google Scholar 

  • Withers, C.S., Nadarajah, S.: Adjusting Cornish-Fisher expansions and confidence intervals for the effect of roundoff. Statistics 46, 627–644 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the Editor and the two referees for careful reading and for their comments which greatly improved the paper.

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Authors and Affiliations

  1. Industrial Research Limited, Lower Hutt, New Zealand

    Christopher S. Withers

  2. University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

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  1. Christopher S. Withers
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  2. Saralees Nadarajah
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Correspondence to Saralees Nadarajah.

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Withers, C.S., Nadarajah, S. Cumulants of a random variable distributed uniformly on the first \(n\) integers. AStA Adv Stat Anal 99, 229–236 (2015). https://doi.org/10.1007/s10182-014-0238-0

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  • DOI: https://doi.org/10.1007/s10182-014-0238-0

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